Question

For Session 17, on the problems and solutions, I'm not understanding how you determine vector OC.

Answer 1

the disk is sliding, which means its center is moving, at the given speed \( 12 \sqrt{2}\) cm/sec in the direction <1,1>

we want a vector \( \vec{OC} \) that represents the position of point C as a function of time "t" (in seconds)

to create such a vector, we first form a vector that points in the direction of travel, and has length equal to the speed that the center is moving (in the given direction)

the easiest way to do make a *unit length* vector in the direction <1,1>
we do this by dividing by the length of <1,1>. The length is \( \sqrt{1^2+1^2}= \sqrt{2}\), and our unit length (direction) vector is
\[ u= \left<\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right> \]
now scale it by the speed to get our *velocity* vector
\[ v= 12 \sqrt{2}\left<\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right> \]

Now if we multiply by time "t" we get the distance moved away from the origin in the direction <1,1>

\[ \vec{OC}= 12 \sqrt{2}t\left<\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right> \\
\vec{OC}= <12t, 12t>
\]

Answer 2

I was struggling to see where the sqrt(2) came from, but that makes sense. Thanks!